Use of the word "data" not in the statistical sense - MathOverflow [closed]

Use of the word "data" not in the statistical sense

2009-12-26T07:57:29Z

Occasionally I see the use of the word "data" in definitions. For instance, one definition of an exact sequence starts off by saying, "An exact sequence of abelian groups (or modules or vector spaces) is given by the data of two homomorphisms [...]" (Perrin, Algebraic Geometry).

I've heard this term used in class as well once. In these instances data of course does not refer to data as used in statistics, as in data from an experiment.

What is the purpose of using such a strange word in abstract mathematics? Has anyone noticed this word?

Answer by Pete L. Clark for Use of the word "data" not in the statistical sense

2009-12-26T08:25:32Z

This use seems in line -- although perhaps not identical -- with the following dictionary definition:

Data:

Factual information, especially information organized for analysis or used to reason or make decisions.
Computer Science Numerical or other information represented in a form suitable for processing by computer.
Values derived from scientific experiments.

It is often used in mathematics in the way you have identified above. Namely, when defining a mathematical structure, it gives the reader a heads up as to the fact that that the structure is "multi-sorted" and involves more than one object. In more formal language, one might say "tuple", e.g.,

"A topological group is a triple $(G,m,\tau)$, where G is a set, $m: G \times G \rightarrow G$ is a binary operation, and $\tau$ is a family of subsets of $G$, such that...."

One could also have said "A topological group is given by the data G,m,$\tau$..."

Answer by David Spivak for Use of the word "data" not in the statistical sense

2009-12-26T15:35:33Z

In mathematics, the word "data" is often used for the most general mathematical noun. One may consider a set, a function, a category, a group, or seven groups "data."

But for data to be interesting, it must have some sort of structure. Thus the data must satisfy certain properties.

A set-with-binary-operation consists only of data: a set S and a function f: S x S ---> S. It becomes a semi-group when we add a requirement to this data: the function must be associative.

In general, people speak of "data" and "structure." These are the raw materials and craftsmanship that create the mathematical universe.